Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{r^3 - 4r^2 - 60r}{-9r^2 - 63r - 54}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {r(r^2 - 4r - 60)} {-9(r^2 + 7r + 6)} $ $ t = -\dfrac{r}{9} \cdot \dfrac{r^2 - 4r - 60}{r^2 + 7r + 6} $ Next factor the numerator and denominator. $ t = - \dfrac{r}{9} \cdot \dfrac{(r + 6)(r - 10)}{(r + 6)(r + 1)}$ Assuming $r \neq -6$ , we can cancel the $r + 6$ $ t = - \dfrac{r}{9} \cdot \dfrac{r - 10}{r + 1}$ Therefore: $ t = \dfrac{ -r(r - 10)}{ 9(r + 1)}$, $r \neq -6$